| 20.10.08, | 14:30 | Evgeny Materov ( Amherst, Berlin) |
| | | Tate resolutions and Weyman complexes
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| | Abstract:
The Tate resolution of a coherent sheaf on projective space is a bi-infinite exact complex over an exterior algebra. The terms of the complex are known from the work
of Eisenbud, Floystad and Schreyer, but the differentials are only partially known. For example, for sheaves arising from Veronese embedding the differentials are
induced by the Bezoutian, for sheaves arising from Segre embedding the toric Jacobian gives the choice of differentials. In my talk I will explain how to use the maps
in Tate resolution to construct the Weyman-style complexes. Weyman complexes are important tools in computation of multidimensional resultants, discriminants and
hyperdeterminants. The lecture is based on joint work with David Cox.
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| 27.10.08, | 14:30 | Nikolai Beck (FU) |
| | | Coset SL(2,R) WZNW Models
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| | Abstract:
The SL(2,R) WZNW model is a classically integrable field theory, whose coset models have interesting properties. Among these models are Liouville theory, which is
important for non-critical string theory, and the SL(2,R)/U(1) model, which can be seen as a toy model for strings in non-trivial background. Due to the integrability
and the rich symmetries of the WZNW model these coset models can be quantized non-perturbatively. In my talk I will introduce the WZNW model, describe how the
SL(2,R)/U(1) coset model is obtained by gaugeing the original action, and finally outline the quantization procedure.
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| 03.11.08, | 14:30 | Jarek Wisniewski (Warsaw) |
| | | Rigidity of Mori cone for Fano manifolds |
| | Abstract:
Last year, during the American Institute of Mathematics workshop on rational curves, a question was raised about rigidity of the Mori cone under deformation of
Fano manifolds (question 0.7 at http://www.aimath.org/WWN/rationalcurves/rationalcurves.pdf). The answer follows easily from my (apparently un-noticed) '91 paper
(Duke Math Journal 64) in which fiber-locus inequality for Fano-Mori contractions is played against Hard Lefschetz Theorem in order to prove rigidity of nef values
(or thresholds). I will recall that old paper and derive some new corollaries.
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| 10.11.08, | 14:30 | Georg Hein (Essen) |
| | | Geometric morphisms to moduli spaces
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| | Abstract: For a smooth projective curve X of genus g, let SU(r)
denote the moduli space of S-equivalence classes of
vector bundles. A morphism Y --> SU(r) is by definition
a geometric morphism, if it is induced by a family of
vector bundles on Y x X. I'll show that dominant
geometric morphisms have even degree at least 2g-4.
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| 17.11.08, | 14:30 | Michael Rapoport (Bonn) |
| | | On Deligne-Lusztig varieties and period domains over finite fields |
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| 24.11.08, | 14:30 | Daniel Hernandez Ruiperez (Salamanca) |
| | | Moduli spaces of sheaves on degenerations of elliptic curves
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| | Abstract: We nd some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional
sheaves. Using them we establish new identications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are
symmetric powers of the curve whilst for positive rank there are only a nite number of non-isomorphic spaces. We prove similar results for the relative semistable
moduli spaces on an arbitrary genus one bration with no conditions either on the base or on the total space. For a cycle of projective lines, we compute all the
stable sheaves of degree 0. Finally, we prove that the connected component of the moduli space that contains vector bundles of rank r is isomorphic to the r-th
symmetric product of the rational curve with one node.
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| 01.12.08, | 14:30 | Herbert Kurke (HU) |
| | | Workshop on: Algebraische Geometrie und die KP-Gleichung"
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| | Vorträge:
1) Keime linearer Differentialoperatoren
2) Schur-Paare und ihre algebro-geomerische Entsprechung
3) Isospektrale Deformationen
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| 08.12.08, | 14:30 | Sergey Galkin (Mainz/Moskau) |
| | | On degenerations of Fanos
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| | Abstract:
We discuss an approach to describe Fano varieties by their degenerations
to special (e.g. toric) singular Fanos and provide a bunch of low-dimensional examples.
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| 15.12.08, | 14:30 | Urs Hackstein (Ulm) |
| | | Principal $G$-bundles on $p$-adic curves and parallel transport
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| | Abstract: We define functorial isomorphisms of parallel transport along \'etale
paths for a class of principal $G$-bundles on a $p$-adic curve. Here
$G$ is a connected reductive algebraic group of finite presentation
and the considered principal bundles are just those with potentially
strongly semistable reduction of degree zero. The constructed
isomorphisms yield continous functors from the \'etale fundamental
groupoid of the given curve to the category of topological spaces with
a simply transitive continous right $G(\mathbb{C}_{p})$-action.
This generalizes a construction in the case of vector bundles on a
$p$-adic curve by Deninger and Werner. It may be viewed as a partial
$p$-adic analogue of the classical theory by Ramanathan of principal
bundles on compact Riemann surfaces, which generalizes the classical
Narasimhan--Seshadri theory of vector bundles on compact Riemann
surfaces.
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| 22.12.08, | 14:30 | Sergey Galkin (Mainz/Moscow) |
| | | On algebraic and arithmetic "quantum" invariants of Fano Varieties
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| 22.12.08, | 14:30 | Vasily Golyshev (Moscow) |
| | | How to discover Fano varieties
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| 05.01.09, | 14:30 | Markus Reineke (Wuppertal) |
| | | "Smooth models of quiver moduli" |
| | Abstract:
Quiver moduli parametrize isomorphism classes of (poly-)stable
representations of quivers up to isomorphism. Analogous to the case of
moduli of vector bundles, there is a distinction between a (numerically
defined) coprime case, with quite well-understood non-singular
projective moduli, and a non-coprime case, leading either to
non-compact, or to highly singular moduli.
The aim of the talk is to formulate and study a closely related moduli
problem, reminiscent of Hilbert schemes, which always produces smooth
projective moduli. Their topology and geometry (in particular, their
Betti numbers) will be described. A relation of smooth models to
Donaldson-Thomas type invariants will be discussed.
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| 12.01.09, | 14:30 | Laura Costa (Barcelona) |
| | | Derived Category of toric varieties with small Picard number
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| | Abstract:
In the talk I will focus the attention to construct a full
strongly exceptional collection of line bundles in the derived
category $D^b(X)$ where $X$ is the blow up of $\PP^{n-r}\times
\PP^r$ along a multilinear subspace $\PP^{n-r-1}\times \PP^{r-1}$
of codimension $2$ of $\PP^{n-r}\times \PP^r$. As a main tool I will
use the splitting of the Frobenius direct image of line bundles on
toric varieties as well as a precise combinatorial description of
acyclic line bundles on toric varieties. Then, I will analyse the existence of full strongly exceptional collections
made up of line bundles on toric varieties with small Picard number.
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| 19.01.09, | 14:30 | Martijn Kool (Oxford) |
| | | Fixed Point Loci of Moduli Spaces of Sheaves on Toric Varieties |
| | Abstract:
Extending work of Klyachko and Perling, we develop a combinatorial
description of pure equivariant sheaves on an arbitrary nonsingular toric
variety X. This combinatorial description can be used to construct very explicit
moduli spaces of stable equivariant sheaves on X using Geometric Invariant
Theory (analogous to techniques used in case of equivariant vector bundles by
Payne and Perling). We show how the moduli spaces of stable equivariant sheaves on X can be used to explicitly compute the fixed point locus of the moduli space of all stable sheaves on X.
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| 26.01.09, | 14:30 | Pawel Sosna (Bonn) |
| | | Derived equivalent conjugate K3 surfaces |
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| 02.02.09, | 14:30 | Andrew Kresch (Zürich) |
| | | Schubert calculus for isotropic Grassmannians
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| | Abstract:
The talk will describe joint work with Anders Buch and Harry Tamvakis
establishing Pieri and Giambelli formulas for the Grassmannians
of subspaces of given dimension, isotropic for a nondegenerate
skew-symmetric bilinear form on a given finite-dimensional vector
space (over the complex numbers). The outcome is an understanding
of how to calculate with Schubert classes on an isotropic Grassmannian.
Such formulas are classical for usual (non-isotropic) Grassmannians,
while in the isotropic setting Pragacz and Ratajski previously gave
a Pieri formula, leaving the Giambelli formula as an open problem.
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| 09.02.09, | 14:30 | Luis Alvarez-Consul (Madrid) |
| | | Coupled equations for Kähler metrics and Yang-Mills connections |
| | Abstract:
In this talk I will study natural partial differential equations
coupling a Kähler metric and a Yang-Mills connection on a vector bundle
over a complex projective manifold. These equations generalise the
constant scalar curvature condition for a Kähler metric and the Hermitian
Yang-Mills equation for a connection. I will provide an algebro-geometric
interpretation for the equations and will study their relationship to a
stability condition associated to the algebro-geometric moduli problem for
pairs formed by a polarised complex variety and a vector bundle. Joint
work with Mario Garcia-Fernandez and Oscar Garcia-Prada.
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| 09.02.09, | 16:30 | Sandra Di Rocco (Sweden) |
| | | Projective duality of toric varieties |
| | Abstract:
Given an embedding of an algebraic variety X in projective
space, the associated dual variety is defined as the Zariski closure
(in dual projective space) of the hyperplanes tangent to X at a non
singular point. For a general embedding the dual variety is a
hypersurface, but there are exceptions, called defective embeddings,
which led algebraic geometers to the problem of classifying them.
This is a very classical problem in Algebraic Geometry, with
important contributions by Kleiman, Ein and many others.
In general, a classification is quite involved and it is known only
up to dimension 10.
In the first part of the talk the general theory of projective
duality will be reviewed. The second part will focus on the
category of toric variety, where Q-factorial defective embedding are
characterized by having the stucture of an elementary toric
fibration. This is a result obtained in collaboration with C.
Casagrande.
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